After Suarez overcame a 5-1 deficit to tie the second set at 5, Stephens broke serve for a 6-5 lead and scored four consecutive points to put away the match, ending it with a sharp cross-court forehand.

"Maybe the old me would have kind of freaked out," Stephens said. "I was kind of like, 'OK, it's fine, she's allowed to play well. Just kind of regroup.' I did that. I was happy with that."

The 25-year-old Stephens, ranked No. 3 in the world, won a hard-court title in Miami in March and lost the French Open final to top-ranked Simona Halep. In the quarterfinals, the American will face Anastasija Sevastova of Latvia, a 6-3, 7-6 (2) winner over 10th-seeded Julia Goerges of Germany.

Sevastova, ranked 19th, won last month in Bucharest.

The top-seeded Halep advanced with a 6-2, 6-2 victory over 13th-seeded Venus Williams in the second match of the day for both players. Halep set up a quarterfinal against sixth-seeded Caroline Garcia of France, a 6-3, 6-2 winner over Russia's Maria Sharapova.